Try to find a family accommodation in one of small bays along the coast like Milna or Ivan dolac etc.
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Hvar town is full of people, bars, music and energy, while stari Grad is calm, has a lovely market with locals selling their produce, a much quieter marina, coffe shops etc. You can always visit the other town with local bus or rent a car - in which case I'd suggest visiting Jelsa, Pitve, Vrisnik (KonobaVrisnik has great food)
and if you're slightzly adventurous, perhaps you might consider driving through a tunnel towards Zavala, then back to Hvar late afternoon on an upaved road that passes through Ivan Dolac and Sveta Nedilja - but only if you are an experienced and confident driver.
depends, if You want quiet surrounding then it is better Stari Grad, for night life city of Hvar is much better. But by car they are 20 minutes apart so You can experience both.